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Machine-Learned Closure of URANS for Stably Stratified Turbulence: Connecting Physical Timescales & Data Hyperparameters of Deep Time-Series Models

Muralikrishnan Gopalakrishnan Meena, Demetri Liousas, Andrew D. Simin, Aditya Kashi, Wesley H. Brewer, James J. Riley, Stephen M. de Bruyn Kops

TL;DR

The paper tackles the challenging problem of closing URANS equations for stably stratified turbulence by learning the entire right-hand side of the reduced-order energy equations with time-series machine learning. It compares LSTM and Neural ODE (NODE) approaches to predict the evolution of energy-like states from DNS data, and introduces a framework to interpret data requirements via learned physical timescales. The authors demonstrate accurate and numerically stable a priori and a posteriori predictions for SST initialized with Taylor–Green vortices and show that the ML-derived timescales relate to the Reynolds and buoyancy-driven flow dynamics, offering a pathway to generalize closures across parameter regimes. The work provides a data-driven, interpretable backbone for SST closure modeling and highlights the potential of time-series models to capture complex turbulence dynamics with reduced computational cost, while outlining future directions in uncertainty quantification and higher-dimensional extensions.

Abstract

We develop time-series machine learning (ML) methods for closure modeling of the Unsteady Reynolds Averaged Navier Stokes (URANS) equations applied to stably stratified turbulence (SST). SST is strongly affected by fine balances between forces and becomes more anisotropic in time for decaying cases. Moreover, there is a limited understanding of the physical phenomena described by some of the terms in the URANS equations. Rather than attempting to model each term separately, it is attractive to explore the capability of machine learning to model groups of terms, i.e., to directly model the force balances. We consider decaying SST which are homogeneous and stably stratified by a uniform density gradient, enabling dimensionality reduction. We consider two time-series ML models: Long Short-Term Memory (LSTM) and Neural Ordinary Differential Equation (NODE). Both models perform accurately and are numerically stable in a posteriori tests. Furthermore, we explore the data requirements of the ML models by extracting physically relevant timescales of the complex system. We find that the ratio of the timescales of the minimum information required by the ML models to accurately capture the dynamics of the SST corresponds to the Reynolds number of the flow. The current framework provides the backbone to explore the capability of such models to capture the dynamics of higher-dimensional complex SST flows.

Machine-Learned Closure of URANS for Stably Stratified Turbulence: Connecting Physical Timescales & Data Hyperparameters of Deep Time-Series Models

TL;DR

The paper tackles the challenging problem of closing URANS equations for stably stratified turbulence by learning the entire right-hand side of the reduced-order energy equations with time-series machine learning. It compares LSTM and Neural ODE (NODE) approaches to predict the evolution of energy-like states from DNS data, and introduces a framework to interpret data requirements via learned physical timescales. The authors demonstrate accurate and numerically stable a priori and a posteriori predictions for SST initialized with Taylor–Green vortices and show that the ML-derived timescales relate to the Reynolds and buoyancy-driven flow dynamics, offering a pathway to generalize closures across parameter regimes. The work provides a data-driven, interpretable backbone for SST closure modeling and highlights the potential of time-series models to capture complex turbulence dynamics with reduced computational cost, while outlining future directions in uncertainty quantification and higher-dimensional extensions.

Abstract

We develop time-series machine learning (ML) methods for closure modeling of the Unsteady Reynolds Averaged Navier Stokes (URANS) equations applied to stably stratified turbulence (SST). SST is strongly affected by fine balances between forces and becomes more anisotropic in time for decaying cases. Moreover, there is a limited understanding of the physical phenomena described by some of the terms in the URANS equations. Rather than attempting to model each term separately, it is attractive to explore the capability of machine learning to model groups of terms, i.e., to directly model the force balances. We consider decaying SST which are homogeneous and stably stratified by a uniform density gradient, enabling dimensionality reduction. We consider two time-series ML models: Long Short-Term Memory (LSTM) and Neural Ordinary Differential Equation (NODE). Both models perform accurately and are numerically stable in a posteriori tests. Furthermore, we explore the data requirements of the ML models by extracting physically relevant timescales of the complex system. We find that the ratio of the timescales of the minimum information required by the ML models to accurately capture the dynamics of the SST corresponds to the Reynolds number of the flow. The current framework provides the backbone to explore the capability of such models to capture the dynamics of higher-dimensional complex SST flows.
Paper Structure (18 sections, 10 equations, 12 figures, 1 table)

This paper contains 18 sections, 10 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Theoretical Background for URANS of SST
  3. Approach
  4. Procedure
  5. Neural network setup
  6. Long Short-Term Memory (LSTM) model
  7. Neural Ordinary Differential Equation (NODE) model
  8. Comparing LSTM and NODE models
  9. Training the models
  10. Results and Discussion
  11. Sample test problem: Simple pendulum
  12. LSTM and NODE results
  13. Interpreting the ML model data requirements: minimum information required by model
  14. SST initialized with Taylor--Green vortices
  15. a priori testing
  16. ...and 3 more sections

Figures (12)

  • Figure 1: Framework of the current modeling approach to predict the time evolution of the reduced-dimensional flow state variables.
  • Figure 2: Illustration of the timescales defined by a sequence in a time-series data.
  • Figure 3: Sample portrayal of the LSTM modeling framework used in the current study for predicting the time evolution of a system of ODEs.
  • Figure 4: Sample portrayal of the NODE modeling framework used in the current study for predicting the time evolution of a system of ODEs.
  • Figure 5: Comparing the time-dependence for the LSTM and NODE models.
  • ...and 7 more figures